Hausman test for panel data

Question

I am performing a Hausman test to decide whether to use fixed effects or random effects model. The results I get are as follows:

. hausman random fixed

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |     random       fixed        Difference          S.E.
-------------+----------------------------------------------------------------
         opm |    .0375723     .0128309        .0247414        .0029346
    tadiv10m |    .0000558     .0000532        2.67e-06        .0000101
   auditcomm |    .0765687     .0572845        .0192842        .0087329
nplsgrossl~n |    .2700572     .0569666        .2130906        .0193822
  govauditor |   -.0628177     -.040186       -.0226318        .0042376
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

chi2(5) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =      341.23
                Prob>chi2 =      0.0000

and

. hausman fixed ., sigmamore

Note: the rank of the differenced variance matrix (4) does not equal the number of coefficients being tested (5);
        be sure this is what you expect, or there may be problems computing the test.  Examine the output of your
        estimators for anything unexpected and possibly consider scaling your variables so that the coefficients
        are on a similar scale.

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |     fixed          .          Difference          S.E.
-------------+----------------------------------------------------------------
         opm |    .0128309     .0375723       -.0247414        .0010828
    tadiv10m |    .0000532     .0000558       -2.67e-06        4.60e-07
   auditcomm |    .0572845     .0765687       -.0192842        .0008345
nplsgrossl~n |    .0569666     .2700572       -.2130906        .0091677
  govauditor |    -.040186    -.0628177        .0226318        .0010184
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =      621.45
                Prob>chi2 =      0.0000

How do I go about interpreting the results?

Answer 1

What you are looking for is the $\chi^2$ statistic produced at the end of the test. The null hypothesis of the Hausman test is that the fixed and random effects model do not differ significantly from each other. A significant test statistic means that we reject the null.

In your case

    Test:  Ho:  difference in coefficients not systematic

chi2(5) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =      341.23
                Prob>chi2 =      0.0000

the value of the test statistic is 341.23 and the p-value shows that this value is significant at the 1% level and even below (Prob>chi2 = 0.0000).

The same holds for your second test but be sure that you pay attention to the note displayed at the top of the test where it warns you that the coefficients between the two models you test are not the same. Stata issues such warnings for good reason normally and choosing to ignore such warnings should be based on a deep understanding of what is going on and that it is a sound decision to ignore the warning (typically it is not).